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Preface | |

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The approximation problem and existence of best approximations | |

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Examples of approximation problems | |

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Approximation in a metric space | |

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Approximation in a normed linear space | |

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The L[subscript p]-norms | |

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A geometric view of best approximations | |

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The uniqueness of best approximations | |

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Convexity conditions | |

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Conditions for the uniqueness of the best approximation | |

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The continuity of best approximation operators | |

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The 1-, 2- and [infinity]-norms | |

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Approximation operators and some approximating functions | |

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Approximation operators | |

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Lebesgue constants | |

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Polynomial approximations to differentiable functions | |

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Piecewise polynomial approximations | |

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Polynomial interpolation | |

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The Lagrange interpolation formula | |

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The error in polynomial interpolation | |

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The Chebyshev interpolation points | |

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The norm of the Lagrange interpolation operator | |

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Divided differences | |

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Basic properties of divided differences | |

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Newton's interpolation method | |

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The recurrence relation for divided differences | |

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Discussion of formulae for polynomial interpolation | |

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Hermite interpolation | |

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The uniform convergence of polynomial approximations | |

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The Weierstrass theorem | |

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Monotone operators | |

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The Bernstein operator | |

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The derivatives of the Bernstein approximations | |

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The theory of minimax approximation | |

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Introduction to minimax approximation | |

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The reduction of the error of a trial approximation | |

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The characterization theorem and the Haar condition | |

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Uniqueness and bounds on the minimax error | |

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The exchange algorithm | |

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Summary of the exchange algorithm | |

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Adjustment of the reference | |

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An example of the iterations of the exchange algorithm | |

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Applications of Chebyshev polynomials to minimax approximation | |

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Minimax approximation on a discrete point set | |

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The convergence of the exchange algorithm | |

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The increase in the levelled reference error | |

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Proof of convergence | |

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Properties of the point that is brought into reference | |

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Second-order convergence | |

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Rational approximation by the exchange algorithm | |

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Best minimax rational approximation | |

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The best approximation on a reference | |

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Some convergence properties of the exchange algorithm | |

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Methods based on linear programming | |

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Least squares approximation | |

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The general form of a linear least squares calculation | |

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The least squares characterization theorem | |

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Methods of calculation | |

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The recurrence relation for orthogonal polynomials | |

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Properties of orthogonal polynomials | |

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Elementary properties | |

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Gaussian quadrature | |

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The characterization of orthogonal polynomials | |

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The operator R[subscript n] | |

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Approximation to periodic functions | |

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Trigonometric polynomials | |

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The Fourier series operator S[subscript n] | |

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The discrete Fourier series operator | |

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Fast Fourier transforms | |

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The theory of best L[subscript 1] approximation | |

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Introduction to best L[subscript 1] approximation | |

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The characterization theorem | |

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Consequences of the Haar condition | |

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The L[subscript 1] interpolation points for algebraic polynomials | |

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An example of L[subscript 1] approximation and the discrete case | |

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A useful example of L[subscript 1] approximation | |

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Jackson's first theorem | |

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Discrete L[subscript 1] approximation | |

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Linear programming methods | |

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The order of convergence of polynomial approximations | |

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Approximations to non-differentiable functions | |

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The Dini-Lipschitz theorem | |

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Some bounds that depend on higher derivatives | |

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Extensions to algebraic polynomials | |

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The uniform boundedness theorem | |

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Preliminary results | |

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Tests for uniform convergence | |

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Application to trigonometric polynomials | |

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Application to algebraic polynomials | |

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Interpolation by piecewise polynomials | |

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Local interpolation methods | |

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Cubic spline interpolation | |

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End conditions for cubic spline interpolation | |

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Interpolating splines of other degrees | |

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B-splines | |

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The parameters of a spline function | |

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The form of B-splines | |

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B-splines as basis functions | |

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A recurrence relation for B-splines | |

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The Schoenberg-Whitney theorem | |

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Convergence properties of spline approximations | |

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Uniform convergence | |

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The order of convergence when f is differentiable | |

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Local spline interpolation | |

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Cubic splines with constant knot spacing | |

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Knot positions and the calculation of spline approximations | |

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The distribution of knots at a singularity | |

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Interpolation for general knots | |

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The approximation of functions to prescribed accuracy | |

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The Peano kernel theorem | |

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The error of a formula for the solution of differential equations | |

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The Peano kernel theorem | |

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Application to divided differences and to polynomial interpolation | |

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Application to cubic spline interpolation | |

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Natural and perfect splines | |

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A variational problem | |

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Properties of natural splines | |

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Perfect splines | |

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Optimal interpolation | |

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The optimal interpolation problem | |

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L[subscript 1] approximation by B-splines | |

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Properties of optimal interpolation | |

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The Haar condition | |

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Related work and references | |

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Index | |